The total external branch length of Beta-coalescents

Abstract

For $1<\alpha <2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-\alpha ,\alpha )$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of ${\tau }_n^{2-\alpha }$ over the entire parameter regime, where ${\tau }_n$ denotes the random number of coalescences that bring the $n$ lineages down to one. This is in contrast to the fluctuation behavior of the total branch length, which exhibits a transition at ${\alpha }_0=(1+\sqrt{5})/2$.

17 pages, 2 figures

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